1.12.3 Disequality

Finally, let us also say something about disequality,
which is negation of equality:11We use “inequality”
to refer to < and ≤. Also, note that this is negation of the propositional identity type.
Of course, it makes no sense to negate judgmental equality ≡, because judgments are not subject to logical operations.

(x≠Ay):≡¬(x=Ay).

If x≠y, we say that x and y are unequal
or not equal.
Just like negation, disequality plays a less important role here than it does in classical
mathematics. For example, we cannot prove that two things are equal by proving that they
are not unequal: that would be an application of the classical law of double negation, see §3.4 (http://planetmath.org/34classicalvsintuitionisticlogic).

Sometimes it is useful to phrase disequality in a positive way. For example,
in Theorem 11.2.4 (http://planetmath.org/1122dedekindrealsarecauchycomplete#Thmprethm1) we shall prove that a real numberx has an inverse if,
and only if, its distance from 0 is positive, which is a stronger requirement than x≠0.